Title
Author(s)
Citation
Issue Date
ON THE COMPLETE CONTINUITY OF OPERATORS IN
AN INTERPOLATION THEOREM
Tetsuya, SHIMOGAKI
Journal of the Faculty of Science, Hokkaido University. Ser. 1,
Mathematics = 北海道大学理学部紀要, 20(3): 109-114
1968
DOI
Doc URL
http://hdl.handle.net/2115/58087
Right
Type
bulletin (article)
Additional
Information
File
Information
JFS_HU_v20n3-109.pdf
Instructions for use
Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP
O NTHECOMPLETECONTINUITYOFOPERATORS
INANINTERPOLATIONTHEOREM
By
TetsuyaSHIMOGAKI
1
. I
nt
h
i
spaper , a Banachs
p
a
c
e (工 1: .11) , whoseelementsa
r
ecomplex
v
a
l
u
ec
1
. Lebesεue m
easurable f
u
n
c
t
i
o
n
s over t
h
ei
n
t
e
r
v
a
l (0 , 1), w
i
l
l be c
a
l
l
e
d
a 品川μch jìmctioll やμcc， i
fi
ts
a
t
i
s
fe
st
h
ef
o
l
l
o
w
i
n
gc
o
n
d
i
t
i
o
n
s
(
l
.1
)
IglζIfI 1 ) ， fEX illψlics 9 モ Xωzd 11σ11ζIlfll;
(1.~)
()~fn ↑ 2) ， I
l
f
"1
1~三 Al
(
7
1= 1, 2 , .一) i
m
p
l
i
c
s
UFz=fcxmdllfil 二 TilIfnilFrom (l.~) i
tf
o
l
l
o
w
st
h
a
tt
h
e norm 1
1
.
11 on a Banach fu 町tion s
p
a
c
ei
s
semicontinuous , i
.e 円。 ζ fn ↑f ， f-fdX i
m
p
l
i
e
si
l
f
i
i
=
y
p
1
i
j
p
til.The space
1
(工 11 .1
¥
)i
sc
a
l
l
e
d Jでarrangcment ùrua バωzt， i
f 0ζfε X i
m
p
l
i
e
s9 ε .
r
-and
I\f\\ 二\\ 9¥ f
o
r each fu町tion g , equimeasurable withf ・ Let V , L∞ be t
h
e
Lebes只ue s
p
a
c
e
sover (0 , 1) , andl
e
t R(V;L∞) bet
h
es
e
to
fa
l
lb
o
u
n
d
e
c
ll
i
n
e
a
r
o
p
e
r
a
t
o
r
sfromeacho
ft
h
es
p
a
c
e
s.L 1, I~∞ into i
t
s
e
l
f
. By ¥
¥
T
l
l
i(i=l , or ニ∞)
wedenotet
h
enormo
fano
p
e
r
a
t
o
rT ε R(V 、 L∞) ont
h
ecorrespondings
p
a
c
e
s
.
For each a>O , J
"i
st
h
e functionεiven by j; , (刈 =f(([.r) ， i
f ([.r<l , f , (.r)=O ,
i
f a.r>l
. W ew
r
i
t
ea
l
s
o
σaf=fa;
(
l
.:
l
)
i
ti
se
a
s
yt
os
e
e that 叫 is ab
o
u
n
c
l
e
c
ll
i
n
e
a
ro
p
e
r
a
t
o
ronX、 if .
Y
"i
sr
e
a
r
r
a
n
g
e
ｭ
a
n
a
n
t
.
ment lIlv
The f
o
l
l
o
w
i
n
g theorem was p
r
o
v
e
c
li
n[
4
]3) :
Theorem A. L
ct ~\' b
e a rcarrangcmc
J
lt illvan山zt B
a
n
a
c
l
zfllJl Ctiοn
s
p
a
c
c
. Thc Jl, for c
v
e
r
yTER(V; J~=) 、 T i
sa b
o
zlIldcdl
i
l
l
c
a
r()ρcrat川- frolJl
1
) 1
/
1 clenote日 t
h
ef
unctio日 cle五 ned b
y1
.
11
I.r) 二
.
1
:
1
1
'
)
'
;
:
;g
(
.
r
)h
o
l
c
l
sa
l
m
o
s
te
v
e
r
y
w
h
e
r
e
.
2)
s
i
m
p
l
y
Wew
r
i
t
e0ζ f;，
↑.
i
f0
'
;
:
;j
i'
;
:
;
}
2,;:; .
1/ し川 1. .r モ (0. 1
)
. ./ζ 9
m
c
a
n
st
h
a
t
l
f()ζ/;ι ↑ itnd p
z
Ul/rι=./ ‘ we w
r
i
t
e O,;:;.t;,•
f
:
1
) T
heorem A was 五 rst p
r
o
v
e
c
lb
y V，ず Orlicz f
o
rO
r
l
i
c
zs
p
a
c
e
s[
7
]
. ^
.P. Calclern
o
rqua日一 linear o
p
e
r
a
t
o
r
si
n[
1
]
. l
n[
4
]T
heorem^i
s
g
a
v
et
h
et
h
e
o
r
e
mi
nfull 日 enerarity f
s
t
a
t
e
df
o
rL
i
p
s
c
h
i
t
zo
p
e
r
a
t
o
r
s
1
1
0
T
. Shilllogα わ
X i
n
t
o itsclf, and
(
1
.4
)
IITllx~IITII ∞・ IIσ"llx 4)
lzolds , WhClで α=IITll ∞・ IITll l l
I
nt
h
i
s paper we d
e
a
l with t
h
e complete c
o
n
t
i
n
u
i
t
yo
f T on _.Y, when
Ti
sc
o
m
p
l
e
t
e
l
yc
o
n
t
i
n
u
o
u
sone
i
t
h
e
rV o
r Lへ and weg
i
v
e an
e
c
e
s
s
a
r
yand
su伍 cient c
o
n
d
i
t
i
o
ni
no
r
d
e
rt
h
a
te
v
e
r
y T ぞ R(V; LW) which i
sc
o
m
p
l
e
t
e
l
y
c
o
n
t
i
n
u
o
u
s on Ll (
o
r 1ア) i
sa
l
s
oc
o
m
p
l
e
t
e
l
yc
o
n
t
i
n
u
o
u
s on X (Theorems 1
and 2
)
.
h
a
tX i
s a Banach f
u
n
c
t
i
o
ns
p
a
c
e which
2
. I
nt
h
e sequel , we assume t
i
sa
l
s
o rearrangement i
n
v
a
r
i
a
n
t
. S
i
n
c
eX i
s rearrangement invariant , X i
s
c
o
n
t
a
i
n
e
di
n I) [5]. 明T e need t
h
e followinεlemma:
Lemma 1.
σ0<α<1 ， 1~11σ"llx~μ1 h
o
l
d
s
.
p1ηof.
The i
n
e
q
u
a
l
i
t
y
: 1~ IIσ"llx i
se
v
i
d
e
n
t
. L
e
t 0~f ε X ， and l
e
t
α = n.
m 1 wherem and n a
r
en
a
t
u
r
a
l numbers with 11<m. S
i
n
c
e σαf= σα
(jX(い))ベ we may assume w
ithout l
o
s
so
fg
e
n
e
r
a
r
i
t
yt
h
a
tfニfχ(O ， (t) ・ Now
we de五 ne σ1 byぁ =Tbισll~f ， 1 宅三 iζ m ， where bi=(i-1).m..l andT勺 i
sat
r
a
n
s
ｭ
:
I
>
/
Z
)(
.
1
:
)=h(x-bん if O<x-b包 <1; (
T
t
)
I
)
(
X
)
=
O
l
a
t
i
o
no
p
e
r
a
t
o
rd
e
f
i
n
e
dby ん: (
o
t
h
e
r
w
i
s
e
. Then gi'-""'g}) , l ，} 二 1 ， 2 , "', rn , and σめ =0 ， i
fi キj.
Put hj 二
乙仇， 1~j ~rn ， where we p
ut g} c = gk "" i
f k>m. Obviously i
tf
o
l
l
o
w
s
t
h
a
tf，-....， hi~hj f
o
ra
l
l i , j , and
I
;h j
smce ση ・ l!l
二 n I; σl~nσ" 川
，f二 σ川， (σ，，1 )= σ川 1σ1'-"'" I: σι ・
，f ，
This
i
m
p
l
i
e
s
刀 IIσ… 'fll=111 戸Ihj|ωllh l 1 = rnIlflい
b
e
c
a
u
s
eX i
srearrangementi
n
v
a
r
i
a
n
t
. Therefore 1Iσ… ，fll~m ・ 71 l
l
l
f
l
lh
o
l
d
s
.
Forana
r
b
i
t
r
a
r
yr
e
a
lα>0 ， t
a
k
en
a
t
u
r
a
lnumbersn , m sucht
h
a
t11 .m 1<α< 1.
W ehavethen IIσJ*117) ~ Ila,,',n 1*11 , her悶 IIσJ*II~m' l1 11If*ll=m.n l
l
l
f
l
l
.
b
t
a
i
n IIσJII"三 IlaJ刊《 αlllfll ， whichp
r
o
v
e
sLemma1
.
L
e
t
t
i
n
g11'm 1 ↑ α ， weo
Nowwe c
o
n
s
i
d
e
rt
h
ef
o
l
l
o
w
i
n
gc
o
n
d
i
t
i
o
n
s on X:
(
2
.
1
)
IIσ"llx< 1
forsomc a>1;
4) I
I
T
l
l
xd
e
n
o
t
e
st
h
enormo
fT o
nX.
5) Xed
e
n
o
t
e
st
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cf
u
n
c
t
i
o
no
ft
h
es
e
te
(
i
) Wew
r
i
t
eJ~g ， i
ffi
se
q
u
i
m
e
a
s
u
r
a
b
l
ew
i
t
hg
7) I
*d
e
n
o
t
e
st
h
eclecreasin只 rearrangement o
f1
f1
(}Jl
Ihe ('0 川ρ/1'1 1' (可OlltiJl llity O{(
)
j
>er
ators i
l
l<
11
1 J
I
I
Il'r
j
)
o
/
<
ll
i
o
l
l Thc()/γ川
(
2
.
2
)
1\ 川 1 ， <υ 1
fο r
111
S
O
J
J
l
l
'0<υ く1.
h
o
l
c
l
sf
o
r everyμ>0 ， a
n
c
l Ilall\\\"~ \\a ,,\\\ ifμ>b>O 、
(
2
.
1
)a
n
c
l(
2
.
2
)a
r
ee
q
u
i
v
a
l
e
n
tt
ot
h
ef
o
l
l
o
w
i
n
gc
o
n
c
l
i
t
i
o
n
sr
e
s
p
e
c
t
i
v
e
l
y
:
Sinceσ(l2= σ11 ・ σ"
(
2
.
3
)
l
i
m¥
l
a
l\\, = 0 、
(
2
.
4
)
lima\\σ，， \\x 士 O.
Lemma 2
. l
fX satiポes t
l
z
ec
o
n
d
i
t
i
o
r
z(2.1) , t
h
e
nlim¥
1T
"1
¥y=0h
o
l
d
sfor
u
c
l
zt
l
z
a
tl
i
m1\ 1'''\\1=0ωzd sup1
¥T
"1\ ∞<∞.
e
v
e
r
ysequ仰 ce {1', J ofR(D;L∞) s
ByTheoremA each T"i
sab
o
u
n
c
l
e
c
ll
i
n
e
a
ro
p
e
r
a
t
o
ronX. For
\Ia，， \lx<ε ・ K 1 where
K=sup\IT"I\oo , since (2.1) i
se
q
u
i
v
a
l
e
n
tt
o(
2
.
3
)
. For such 寺 >0 ， we choose
Pnοof
anyε>0 we can 五 nd an 可> 1 s
uch t
h
a
t a> 可 implies
7
l
:
2
:1
an l
l
os
ol
a
r
g
et
h
a
tK
¥
I
T
"
I
¥
l
l
>;
rh
o
l
c
l
sf
o
r each 1l 2 1l o ・
g
e
tf
o
r n 二三 n。
\I 1'，，\\\~ \\7~， し， \\σ rtn\\X'
Then ， by (
l
.4
) we
μι" 二 \\T，，\\
S
i
n
c
e σ~ 二 σ~σ"" h
o
l
d
s with b" 士 \\1'，，\\ ∞ K 1
~1a
n
c
l c" 二 K.\\T"\\l\ i
t
f
o
l
l
o
w
s from Lemma l thatlhJly < iMfJ
o
b
t
a
i
nf
o
r 112110
Ill~， I\Y~ K \Ia"nllx~ε ラ
which completes t
h
e proo
f
.
An o
p
e
r
a
t
o
r1
1on /}i
sc
a
l
l
e
danavemging(~ρerator， i
fA i
sc
l
e
fn
e
dby
(
2
.
5
)
/lf 二九円円卜点 d(ei) l(Lf(.r) ル )χcい
where e i ηeJz 件、 i
f i キj ， Ueic(O 弓 1), and Jl ニ 1 ， 2 ，
・・・
The averaginξ
o
p
e
r
ｭ
'i
L
1
;L ∞) c
l
e
a
r
l
y
. WhenX
srearrangementinvariant , both
a
t
o
r
sbelongt
o B(
r
e always c
o
n
t
r
a
c
t
i
o
n
s on X , becal脱 f '，> ベr andf'(>f-Af
A and 1-A a
h
o
l
d
. MoreoverA i
sw
e
l
lknown , t
h
e
r
e
sc
o
m
p
l
e
t
e
l
ycontinuouson X. As i
h
a
t.
e
x
i
s
t
sasequenceo
c
o
n
v
e
r
g
e
s
stro
ly
fa\ア eragl 昭 operators {11 ,,} sucht
昭
1
"
t
o1, t
h
ei
c
l
e
n
t
i
t
yoperator , on V. Nowwecanprovet
h
ef
o
l
l
o
w
i
n
gtheorem:
， zgement i
n
v
a
r
i
a
n
t
. l
h
a
tevelツ
n ordert
etX b
e rearrω
Theorem 1
. L
1
TEB(D; /~∞) w
l
z
i
c
l
zi
sc
o
m
p
l
e
t
c
l
yc
o
n
t
i
n
u
o
l
l
s011 L b
ca
l
s
o COlllρ letely C07 ト
tinuous ο 11 X , i
2
.
1
)
.
tis 町cessωツ and Sl~t声 α・ellt t
h
a
tX satiポes (
H)
.1> 。… s
t
h
a
t
foγall
tf *( 川三r:川市
p
a
c
c
B
a
n
a
c
hf
u
n
c
t
i
o
ns
Xi メ日 rcarrangcment ln\' 江口日 nt
.
O<t<l
t> ロ川 ('s 11/11 三 Ilgll.
i
f
T
. Shi/ll oまはわ
1
1
2
L
e
t V1be a u
n
i
tb
a
l
lo
f V. By t
h
eassumption
1
'V1 i
sc
o
n
t
a
i
n
e
di
nacompacts
e
to
f V. Hence , fo> r asequenceo
fa
v
e
r
a
g
i
n
g
operat旬
or目s {い
11ηJ
小}， we1
作'Oof
S/.~fficiency.
η →∞
fEV
κηι →白
t
o
g
e
t
h
e
r with 1
1
(1-A ,,) 1' 11 ∞:Ç 11 1' 11= , n 二三l.
1
tf
o
l
l
o
w
s from Lemma 2 t
h
a
t
limll(I-A ,JTllx=O. S
i
n
c
e each An 1', 112
:1i
sc
o
m
p
l
e
t
e
l
ycontinuous onX ,
l
'i
sa
l
s
oc
o
m
p
l
e
t
e
l
yc
o
n
t
i
n
u
o
u
s on X.
Nccessiか
Suppose t
h
a
t 11σ"llx= 1 f
o
ra
l
l a>l
. Then for 正l" =2 2η we
canf nd
anσn EX， gn 二三 o 、 andI
l
g
"
l
l
=
lsucht
h
a
tI
l
a
"
.
.
g
"
I
I
>
!
.P
u
t
t
i
n
g g;， = σバ" ,
we have
n=1 ， 2 ，
11σ2叩;， 11> き and 11σ:， II :Ç 1,
・
S
i
n
c
eX i
srearrangementinvariant , wemayassumewithoutl
o
s
so
fg
e
n
e
r
a
r
i
t
y
t
h
a
t g:, =g;,X" , where χ" i
st
h
ec
h
a
r
a
c
t
e
r
i
s
t
i
cf
u
n
c
t
i
o
no
ft
h
ei
n
t
e
r
v
a
l
: rη 二
(
2'
¥:2 η1). Moreover , by t
h
es
e
m
i
c
o
n
t
i
n
u
i
t
yo
ft
h
e norm 1
1
.1
1 we may
assume t
h
a
t g;, i
sas
i
m
p
l
ef
u
n
c
t
i
o
nf
o
r every 71 2
:1. Now l
e
t σn
昨日人."】 ('n ， 山 where 正t川三 o a
n
c
l2 "ω <C" ， I<"'<C川:Ç 2 ,, 1
A"d
e
ｭ
h
ei
n
t
e
r
v
a
l
s1
"
" = (c川 l' Cn ， よ ν=
n
o
t
e
st
h
ea
v
e
r
a
g
i
n
go
p
e
r
a
t
o
r cle五町cl by t
1, 2 , "'， m 川 that is ,
A，，1オ (Cn ← C"". )
1l(t
ル) d
x
)
X(c""
,,",,
jε/)
.
P
u
t
t
i
n
g T" 二 σ2η A" ， we have f
o
r every n 二三 1 a l
i
n
e
a
ro
p
e
r
a
t
o
r1
'
" belonεmg
t
o R(V;L∞) with I
I1
'
1
=2 n and 11 1'，， 11 ∞= 1
. S
i
n
c
e 1',,1= 1',,(fXn ) an
n1
1
1'，，1=(1'，，1) χ J n ， ，Jll =(2- 2 ヘ
22
"1
) hold f
o
ra
l
l n 二三 l ， t
h
eo
p
e
r
a
t
o
r T =211L
I
i
sdefì配d onL∞ also and1
1TII ∞=l.
On t
h
eo
t
h
e
r hand , a
sI
I
T
I
1
1:ﾇ.6 111~， 111
=1 , l
'a
c
t
sa
l
s
o from ]} i
n
t
oi
t
s
e
lf
. Furthermore as an operator 011 J) , l'
i
sc
o
m
p
l
e
t
e
l
y continuous , a
si
se
a
s
i
l
ys
e
e
n
. The o
p
e
r
a
t
o
rT t
h
u
s defìned ,
however , i
sn
o
tc
o
m
p
l
e
t
e
l
y continuous a
s an o
p
e
r
a
t
o
r on .
Y. 1
n fact , f
o
r
1Tg;,1
1>きIf t
h
e sequence
each Jl 三 1 ， 1'g;, = 7~， g;， 二 σ2 n !l llg;L = σ2 ，.g;" hence 1
{T ι} c
o
n
t
a
i
n
sasubseq配即e whichconvergesi
nt
h
enorm1
1.
1
1t
oanelement
h
el
i
m
i
t mustbe 0 , s
i
n
c
e l' 叫 converges t
o 0 almost everywhere by
o
fX , t
v
i
r
t
u
eo
f(
l
.2
)
. This i
s ac
o
n
t
r
a
c
l
i
c
t
i
o
n
. Thus t
h
en
e
c
e
s
s
i
t
yo
ft
h
ec
o
n
d
i
t
i
o
n
(
2
.1
)i
sp
r
o
v
e
d
.
I
f l'ε B(V; L∞) i
sc
o
m
p
l
e
t
e
l
y continuous on L= , t
h
es
e
t 1' V，∞ IS c
o
n
ｭ
ti
sseparable , where V∞ is au
n
i
tb
a
l
l
t
a
i
n
e
di
n acompacts
e
to
f Lぺ hence i
o
f I~∞ Then ， a
si
sw
e
l
lknown , t
h
e
r
ee
x
i
s
t
sasequenceo
fa
v
e
r
a
g
i
n
go
p
e
r
a
t
o
r
s
{
!
1
n
} sucht
h
a
tAnconvergest
o1s
t
r
o
n
g
l
yon 1' Voo ・ As s
i
m
i
l
a
r
l
ya
sLemma
(
)
I
I/
1
11' ('olllj， ll'/1' 仁川ltilllli/y oj οj'l'ra/ors i
l
l(
/
I
I 111ftソアola/ iOI
l Th
l
'
orclIl
1
1
3
2wecanprovet
h
a
tboth l
i
mV
I1
'
"II~ 二 o a
n
c
lsup11 1',, 11\ <∞ imply l
i
m\1 '1',-1 1, =0
l-'=
n 二と I
η
》∞
p
r
o
v
i
c
l
e
dt
h
a
t X satisfìe自 (2.2). On t
h
eo
t
h
e
r hand , i
fX v
i
o
l
a
t
e
s (2.2) 、 we
canc
o
n
s
t
r
u
c
tano
p
e
r
a
t
o
rl
'o
f R (V; IJ∞) which i
scompletelycontinuouson
Lぺ but n
o
t on ,Y. Such an o
p
e
r
a
t
o
r can be c
o
n
s
t
r
u
c
t
e
c
li
n as
i
m
i
l
a
r way
9)
a
si
n Theorem 1
. Thus we g
e
t
Theorem 2
. LetX b
e 7でarrangement i
n
v
a
r
i
a
n
t
. I
I
I 0パler t
h
a
teue7ツ
Tε B(V;L∞) wh
i
c
l
zi
s comρletely c
O
l
l
t
i
n
u
o
u
sO!l L∞ be (
[
l
s
ocotr.ψletely C
OJlｭ
t
i
n
u
o
u
son X , i
ti
s ne印刷1ツ and s
u
f
f
i
c
i
e
n
tt
h
a
t X sati巴~fies (
2
.
2
)
.
3
. I
nt
h
i
ss
e
c
t
i
o
n we g
i
v
e as
i
m
p
l
ec
o
n
c
l
i
t
i
o
ne
q
u
i
v
a
l
e
n
t with (
2
.
1
)o
r
(2.2) , when _yi
s one o
fsomec
o
n
c
r
e
t
es
p
a
c
e
s
: O
r
l
i
c
z spaces , Lorentzs
p
a
c
e
s
A(cp) , and A1(判 [2]. In [
6
]i
ti
s shown t
h
a
tt
h
ec
o
n
c
l
i
t
i
o
n(
2
.
2
)i
se
q
u
i
v
a
l
e
n
t
t
ot
h
ep
r
o
p
e
r
t
yt
h
a
t XEHLP , i
.e. ，大 X implies 句fE X , where(
)
fi
st
h
e品川ly
Littlewoodmajoranto
ff
. A necessaryanclsu伍 cient conditionfor the conｭ
d
i
t
i
o
n(
2
.
2
)i
sa
l
s
o given i
n [3 , 6
]f
o
rO
r
l
i
c
z spaces , o
rs
p
a
c
e
s/
1(判
For
a Banach f
u
n
c
t
i
o
ns
p
a
c
e .X we denote by X t
h
e COfυiugate s，ρμCC of"""", t
h
e
s
e
to
fa
l
l 凶明ue
al叩 X.
measur枇
The 叫
Ilfll侭l同
吋《乳寸
ρ
阿刈
1} 俳孟)
孟 iおS
f
u
n
c
t
i
o
n
sσ
吋
t
h
a
t)
:
〉川|げ
1ν
刊
If(t
j点列州州{は同
t討)引
州州
W
t約州川)川|
…
p
a
i
rX an
町
n吋
d X, s
i
n
c
e正αz- 礼
1σ
仇叫" ' i
st
h
econ司
o
ft
h
e operatorσα ・ As L N=IJx , where N i
st
h
e complementary
b
t
a
i
n by [
3
; Theorem 4 , o
r6
; Theorem :
l
]
f
u
n
c
t
i
o
no
f ]\,;1, we o
(υ2.1)川
an
凶
1叶
d (υ2.2
勾) ar配e mu
叫
1此旬
tl山
ua
氾
all
勘1)匂
ycl
山
uaι1 白
f or日th
恥
e
JU伊te
Theorem 3
. i
) LM s
a
t
i
s
f
i
e
s(
2
.
1
) if ω1([ only ザ A1 sμ tiポes t
h
e.
:
12e. , tJZC7で c.rist u o :2: 0 ωzd 7>0 s
u
c
ht
h
a
t 1\釘 2u)~7M(u) fora
l
l
conditio刀、 i.
zl 二三 U o
i
i
) L ,!f Sμtiポω(2.2) ザ ωzd only ザ 1"1 satiゆies t
h
e.:1 2 ・ condition.
Fo
rt
h
es
p
a
c
e
s/
1(cp
判外)， Put φ釦釦叫
川
(υ
I刈)=)>(帆 0<ぷパ
ω<1
non
町decreas幻Il1宗 C
∞
O町ave f
u
n
c
t
i
o
n on (川0 ， 町1).
φ釦(什
μT吋噌 おi s
)
a 似p卯附問阿
削叫】洛削削
oω
S臼1此叫t竹
I
n [3 , 6
]i
ti
s shown t
h
a
t(
2
.
2
)i
s
e
q
u
i
v
a
l
e
n
tt
o
(
3
.1
)
l
i
msupφ (2u) φ (u) 1<2.
品→ O
On t
h
eo
t
h
e
r hand , we can prove t
h
a
t(
2
.
1
)i
se
q
u
i
v
a
l
e
n
tt
o
(
3
.
2
)
l
i
miぱ
φ (2u) φ(u)
1>1.
fU
3
.
2
)i
s true , then φ (2u) φ (u) 1
:
:
:
:
;1 十 å ， u~uo<l f
o
rsome
I
nfact , i
9) M
a
k
i
n
gu
s
eo
ft
h
e faιt t
h
a
t rJ a
n
da lrJ([
p
r
o
v
eThcorcm2
f
l
1
Oand
a
r
en
l
u
t
u
a
l
l
yconju 日 ate. we c
a
na
l
s
o
1
1
4
'1 、
>O.
1
I0
Shilllogol"
Put α= :2 .
1
I0 1
. Then f
o
r any a with (
)<μ く 1 we havヒ 101
\1σαx 1o ，'l) I\'1 = 11χ10 ・ 2 11Vt)I11 ミミ (1 十ぷ) 111χ( い，'1 )\\1<(1 十 ò) 1\\ χ(04)11 ・
Thisi
m
p
l
i
e
s l\aJ\11'"ミ (1 十 ò) l
l
l
f
¥
¥1f
o
ra
l
lfε イ (\D)， whichshows 1\σ "\\1<(1 + ) 1
. Hence (
2
.
1
)h
o
l
d
s
.
<1
Conversely , i
f(
3
.
2
) does not hold , we can 五 nd sequences o
fp
o
s
i
t
i
v
e
numbers {ι} and {e ,J
such t
h
a
t μ，， <2 ヘ叫ι ↓， φ (2a ，，) φ (a ，，) 1
:
:
;
;1 十 ε n' and
Let b" 二 2 Jl a n and χ n = X(O ，fJ n ) ・ SinceφIS a conｭ
εη = (
n
2
"
) 1for every 1
1~三1.
cave function , φ(2勺，，) <(1
+2"e ，，) φ(ι) ，
\\χ"\\.111σ24Jl f1 二 φ(2勺μ)φ(μ，，) 1<(1 十
11~1 ， andlimsupIlanl\.l~ l.
2 11 s n
n 二三 1 h
o
l
d
s
. 1
tfollowsfrom t
h
i
st
h
a
t
)二 1 斗 n 1
He即e \\σ2" \ll~三 (1 十 n 1
)1
Thisis , however , i
n
c
o
n
s
i
s
t
e
n
twith(
2
.
:
1
)
. There司
fore we have
Theorem 4
. i
) .
1
(¥
D
)s
a
t
i
s
f
i
c
s (:2 .1) グ ωzd o1l1
y if φ satisfics U
l
.
2
)
.
i
i
) 1
1(
¥
D
)s
a
t
i
s
f
i
c
s(2.2) if ωzd οnly if φ satls斤:cs (
3
.
1
)
Since the spaces 1
1(判 ancl M(判 are mutually conjugate [2 ], we obtain
immecliately from Theorem 4
Theorem 5
. i
) A
l(¥
D
) satisfù吋 (2.1) ~f ωzd οnly if φ satisfil's (
3
.
1
)
.
i
i
) M(
¥
D
)satis戸内 (2.2) {/ωld only if φ s
a
t
i
.
.
買がれ (
3
.
2
)1
1
)
.
References
l
/
l
' th{'ort'lIl
[1
] A
.P
.Cl\LDERδN: Sj>(l lγ ，1>1'1< "('('11 Llolld /，白 alld t
S
t
u
d
i
aMath. ，立(i (19(治)， 2
7
:
1
2
9
9
[
2
] C;. C;. LOREN'I Z: ο11 Ihe theo/~y nj
'sρ山川 1 ， Paci五 c J
. Math 司
‘
.f
[3
:]
1'10jO J'(l1lぉ 1 /1 ゆa Clザ of i
l
l
ll'grabl
l
' imclio /l s ,
υj'
Jlarcillki l'7c;c::: ,
1 (1951) 、 411 4
2
9
Amer
.J
. Math. , 7
7
(1955) , 4
8
4
4
9
2
[
!
¥J (
;
.C;. LORENTZa
n
dT. SHIMOGAKI: J1ajo/'al
l
l
sf
.
疵
illtcrj川 laliο11 tl"，o/γIIIS， t
h
e
)
r
.A
nandaRaur(
i
np
r
i
n
t
i
n
g
)
M
e
m
o
r
i
a
lVolume1
0]
[
5
] E
.M
.SEMENOV: 1111;'ιldillg thcorl'l/l:、 .fòr Ball 山 h .filllctiο11 .\þ山川 υ 1111 山口 IIra !>l1'
fÚllctiοIIS， Do
k
l
.A
k
a
d
.NaukSSS R, 1
5
( (19f洲、 1292-1295 (Russian) ー
[(
ﾌ
] T
. SHIMOGAKI ・ ]1ardy-Littlnc'ood 川匂川-{[II t
si
/
l .f川ICtlOll
，'Ìρ 山山， .
1M
a
t
h
.S
o
c.
.
]
a
p
a
n1
7(1965) , 3
(5
3
:
7
:
i
.
[
7] W. ORLICZ: ο/[ a dasバザィ小川，(It川川口 'Z'(T thc 、/川 Cl' o
/
' illt l'gndル jú /
[
(
'
/i
OI
l.1',
S
t
u
d
i
aMath. , 1
4(1951) , :
l
O2
3
0
9
D
e
p
a
r
t
m
e
n
lo
fM
a
t
h
e
m
a
t
i
c
s
HokkaidoU
n
i
v
c
r
s
i
t
v
(
R
e
c
e
i
v
c
d Apri l, 20 ,
1叩
0)
The
…
nor口1
t
1
1.11/川
lof
山
f
χ (刊0 ，〆川川
F“t川川川
)山;1，1= φ (
υ)
1
1
) 1
n[:い問
i河] acωm
…
叩n
0
町
川出
dliti
凶川
耐州仙
川凶
It山](
ti
somcfx
c
dcon 日 tant "1>0
企" 1φ (a
刈)， ()く μ<1 、 for
19(治)
© Copyright 2024 ExpyDoc
